Monitoring voltage stability of a transmission corridor

ABSTRACT

A voltage stability monitoring apparatus monitors the voltage stability of a transmission corridor through which power flows between different parts of a power system. The apparatus monitors an equivalent load impedance at an interface between the transmission corridor and a part of the power system designated as generating the power. This equivalent load impedance at the interface comprises a ratio of a voltage phasor at the interface to a current phasor at the interface. The apparatus tracks a Thevenin equivalent voltage and impedance of the designated part by separately updating that voltage and impedance. Notably, the apparatus updates the Thevenin equivalent voltage to reflect the magnitude of any changes in the voltage phasor that are associated with large variations in the magnitude of the equivalent load impedance at the interface. The apparatus computes an index indicating the voltage stability as a function of this tracked Thevenin equivalent voltage and impedance.

RELATED APPLICATIONS

The present application claims benefit of U.S. Provisional Application61/825,121, filed May 20, 2013, the disclosure of which is incorporatedherein by reference in its entirety.

TECHNICAL FIELD

The present invention relates generally to monitoring the voltagestability of a transmission corridor through which power flows betweendifferent parts of a power system, and in particular relates tocomputing an index indicating such voltage stability.

BACKGROUND

A power system generates electric power at one part of the system andtransmits that power via a transmission corridor for use by another partof the system. The transmission corridor is considered to be stable interms of voltage if the corridor maintains steady acceptable voltagesnot only under normal operating conditions but also after a disturbanceto the system (e.g., a line outage). A voltage stable corridor thereforeregains acceptable voltages after a disturbance, rather than oscillatingor monotonically decreasing even in response to attempted voltagerestoration mechanisms.

Transmission corridor voltage instability causes power blackouts andtherefore has huge economic and societal costs. Known approaches topreventing blackouts monitor the corridor's real-time proximity tovoltage instability and take appropriate control and protective actionsas needed to mitigate system degradation or disturbance propagation. Forexample, such actions may include load shedding.

Many of these known approaches exploit the relation of the corridor'svoltage instability to the power system's maximum loadability. Inparticular, the approaches identify the corridor's voltage instabilityas being strongly related to the inability of the combined generationand transmission parts of the system to provide the power requested bythe receiving (i.e., load) part of the system. The approaches thereforeemploy a voltage instability criterion expressed directly or indirectlyin terms of the system's maximum deliverable power, which is reachedwhen the magnitude of the Thevenin equivalent impedance of the combinedgeneration and transmission parts of the system equals the magnitude ofthe equivalent load impedance of the power receiving part: |Z _(Th)|=|Z_(l)|.

At least some of these approaches estimate the Thevenin equivalentimpedance of the combined generation and transmission parts of thesystem in stages. Such multi-stage approaches involve estimating thepower generating part's Thevenin equivalent. Some known techniques for“estimating” the power generating part's Thevenin equivalent simplyassume that either the Thevenin equivalent voltage or impedance of thepower generating part is known. See U.S. Pat. No. 7,200,500 B2, April2007, which is incorporated by reference herein in its entirety. Othertechniques actually identify (i.e., track) the power generating part'sThevenin equivalent in the interest of accuracy, i.e., without makingthe above assumption. One such tracking technique performs recursiveleast squares using voltage and current phasor measurements taken at aninterface between the power generating part and the transmissioncorridor. See U.S. Pat. No. 6,219,591 B1, which is incorporated byreference herein in its entirety. To avoid delays associated with therecursive least squares technique, an alternative tracking techniqueseparately updates the power generating part's Thevenin equivalentvoltage and impedance. S. Corsi and G. N. Taranto, “A real-time voltageinstability identification algorithm based on local phasormeasurements,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1271-1279,August 2008.

SUMMARY

One or more embodiments herein track the Thevenin equivalent of a powergenerating part of a power system with improved accuracy as compared toknown tracking techniques. This improved accuracy advantageouslyprevents or at least mitigates false alarms in terms of prematurelydetecting transmission corridor voltage instability.

More particularly, embodiments herein include a method of monitoringvoltage stability of a transmission corridor through which power flowsbetween different parts of a power system. The method is implemented bya voltage stability monitoring apparatus. The method includes monitoringan equivalent load impedance at an interface between the transmissioncorridor and a part of the power system designated as generating thepower. The equivalent load impedance at this “power generating part”interface comprises a ratio of a voltage phasor at the interface to acurrent phasor at the interface.

The method further includes tracking a Thevenin equivalent voltage andimpedance of the designated part by separately updating that voltage andimpedance. Updating the Thevenin equivalent voltage in this regardcomprises updating the voltage to reflect the magnitude of any changesin the voltage phasor that are associated with large variations in themagnitude of the equivalent load impedance at the power generating partinterface. Such large variations include variations greater than athreshold-defined variation.

The method finally includes computing an index indicating the voltagestability as a function of the tracked Thevenin equivalent voltage andimpedance.

In at least some embodiments, updating the Thevenin equivalent voltagecomprises, for each of a plurality of phasor measurement times,determining whether or not variation in the magnitude of the equivalentload impedance at the power generating part interface since a previousphasor measurement time is greater than the threshold-defined variation.If so, the method comprises adjusting the Thevenin equivalent voltagecomputed for the previous phasor measurement time by the magnitude ofthe change in the voltage phasor since that previous phasor measurementtime.

Thevenin equivalent voltage in some embodiments is also updatedresponsive to small variations in the magnitude of the equivalent loadimpedance at the power generating part interface. Specifically, theThevenin equivalent voltage is decreased or increased by a predefinedpercentage change when those small variations do or do not have the samepolarity as variations in the Thevenin equivalent impedance,respectively. Such small variations include variations less than thethreshold-defined variation.

In some embodiments, updating the Thevenin equivalent voltage comprisesupdating the Thevenin equivalent voltage's complex value in rectangularcoordinates. Alternatively or additionally, the method includesdynamically adjusting a threshold defining the threshold-definedvariation, as a function of the Thevenin equivalent voltage.

Additionally, in one or more embodiments, updating the Theveninequivalent impedance comprises solving a set of two linear equationswith two unknown variables. These two unknown variables comprise thereal and imaginary parts of the Thevenin equivalent impedance. Knownvariables in the set of two linear equations include the real andimaginary parts of the Thevenin equivalent voltage, as updated toreflect the magnitude of any changes in the voltage phasor at the powergenerating part interface.

Still further, the method in some embodiments entails dynamicallyadapting which part of the power system is designated as generating thepower. Such dynamic adaptation is performed responsive to detecting achange in direction or magnitude of power flowing through one or bothinterfaces between the transmission corridor and the parts of the powersystem.

Finally, in one or more embodiments, the method involves monitoringwhether a breaker for each line associated with the power generatingpart interface is open or closed. In this case, monitoring theequivalent load impedance at that interface comprises dynamicallycomputing the equivalent load impedance at the interface exclusivelyfrom phasor measurements taken at lines whose breakers are closed. Insome of these embodiments, responsive to detecting the opening orclosing of one or more of these breakers, the method entails updatingthe Thevenin equivalent voltage to reflect the magnitude of theresulting change in the voltage phasor, as dynamically computed, withoutre-initializing the Thevenin equivalent voltage.

Embodiments herein further include a voltage stability monitoringapparatus configured to perform the above-described method.

Of course, the present invention is not limited to the above featuresand advantages. Indeed, those skilled in the art will recognizeadditional features and advantages upon reading the following detaileddescription, and upon viewing the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a power system and a voltage stabilitymonitoring apparatus according to one or more embodiments.

FIG. 2 is a logic flow diagram of a method implemented by a voltagestability monitoring apparatus according to one or more embodiments.

FIG. 3 is a schematic diagram of a circuit model of the power systemaccording to one or more embodiments.

FIG. 4 is a logic flow diagram of a process for tracking the equivalentload impedance at the power generating part interface and for updatingthe Thevenin equivalent voltage at that interface, according to one ormore embodiments.

FIG. 5 is a schematic diagram of a power system circuit model whichillustrates the equivalent load impedance at the power generating partinterface, according to one or more embodiments.

FIG. 6 is a schematic diagram of a power system circuit model whichillustrates modeling the transmission corridor as a T-equivalentcircuit, according to one or more embodiments.

FIG. 7 is a schematic diagram of a power system circuit model whichillustrates modeling the combined generation-transmission parts of thesystem as a Thevenin equivalent circuit, according to one or moreembodiments.

FIG. 8 shows an exemplary transmission corridor for accounting forstructural changes within the corridor, according to one or moreembodiments.

FIG. 9 shows exemplary scenarios for dynamically adapting which part ofthe power system is designated as the power generating part as afunction of aggregate active power, according to one or moreembodiments.

FIGS. 10(a)-10(c) illustrate quantitative advantages of accounting forstructural changes within the corridor, according to one or moreembodiments.

FIGS. 11(a)-11(c) illustrate quantitative advantages of dynamicallyadapting which part is designated as the power generating part,according to one or more embodiments.

DETAILED DESCRIPTION

FIG. 1 illustrates a power system 2 according to one or moreembodiments. The power system 2 generates electric power at one part ofthe system 2 and transmits that power via one or more transmission lines4 for use by another part of the system 2. As shown, for example, partsA, B, and C of the power system 2 are each characterized by some degreeof power generation (G) and loading (L). But power in the aggregateflows in a direction from part A towards part B via the one or moretransmission lines 4. Regardless of the actual power flow direction,though, such characterization of the power system 2 into different partseffectively means that different transfer cuts A and B define a(virtual) transmission corridor 6 between those parts.

The transmission corridor 6 is composed of the one or more physicaltransmission lines 4 via which power flows between the power systemparts. FIG. 1 shows these transmission lines 4 interface parts A and Bof the power system 2 at physical buses A1-A2 and B1-B3. This means thatthe corridor 6 interfaces parts A and B at interfaces Int_(A) andInt_(B). As shown, interface Int_(A) is a virtual bus comprising a groupof physical buses A1-A2. Likewise, interface Int_(B) as shown is avirtual bus comprising a group of physical buses B1-B3.

The transmission corridor 6 is considered to be stable in terms ofvoltage if the corridor 6 maintains steady acceptable voltages not onlyunder normal operating conditions but also after a disturbance to thesystem 2 (e.g., a line outage). A voltage stable corridor thereforeregains acceptable voltages after a disturbance, rather than oscillatingor monotonically decreasing even in response to attempted voltagerestoration mechanisms.

A voltage stability monitoring apparatus 8 is configured to monitor thevoltage stability of the transmission corridor 6, in order to detect inreal-time the proximity of the corridor 6 to voltage instability. Theapparatus 8 in this regard comprises one or more communication interfacecircuits 10 configured to communicatively couple the apparatus 8 to aplurality of time synchronized phasor measurement units (PMUs) 12deployed at both ends of the corridor 6. Any given PMU 8 deployed at aninterface between the corridor 6 and a part of the system 2 measures avoltage phasor and/or a current phasor locally at that interface, andcommunicates those phasor measurements to the apparatus 8 (e.g., at arate of 10-120 samples per second). For example, PMUs 12 deployed atInt_(A) measure voltage phasors at and current phasors in transmissionlines 4 associated with physical buses A1-A2.

The voltage stability monitoring apparatus 8 also comprises one or moreprocessing circuits 14 configured to use the received phasormeasurements to compute an index indicating the corridor's voltagestability. The one or more processing circuits 14 do so according to theprocessing 100 illustrated in FIG. 2.

As shown in FIG. 2, processing 100 by the one or more processingcircuits 14 includes monitoring an equivalent load impedance at aninterface Int_(g) between the transmission corridor 6 and a part of thepower system designated as generating said power (e.g., Int_(A) inFIG. 1) (Block 110). This interface Int_(g) is also referred to hereinas the “power generating part interface” for convenience. The equivalentload impedance at this interface Int_(g) comprises a ratio of a voltagephasor at the interface Int_(g) to a current phasor at the interfaceInt_(g). In at least some embodiments, these voltage and current phasorsat the interface Int_(g) are computed from the voltage and currentphasor measurements received from the one or more PMUs 12 deployed atthe one or more physical buses which collectively form that interfaceInt_(g). Regardless, the equivalent load impedance at interface Int_(g)as used herein characterizes the load “seen” by interface Int_(g) andtherefore characterizes both the transmission corridor 6 and the powerreceiving part of the system 2.

Processing 100 by the one or more processing circuits 14 furtherincludes tracking a Thevenin equivalent voltage and impedance of thedesignated power generating part, by separately updating that voltageand impedance (Block 120). As used herein, separately updating theThevenin equivalent voltage and impedance means updating the Theveninequivalent voltage separately from updating the Thevenin equivalentimpedance, rather than jointly updating both the voltage and impedanceby simultaneously solving for them. Notably, updating the Theveninequivalent voltage comprises updating the voltage to reflect themagnitude of any changes in the voltage phasor at the interface Int_(g)that are associated with large variations in the magnitude of theequivalent load impedance at the interface Int_(g). Large variations inthis regard include variations greater than a threshold-definedvariation.

Processing 100 finally includes computing an index indicating thecorridor's voltage stability as a function of the tracked Theveninequivalent voltage and impedance (Block 130). This index as used hereincomprises any indicator that quantifies the transmission corridor'sproximity to voltage instability. With the corridor's voltage stabilityquantified in this way, actions can be taken as needed to control thecorridor's stability and/or mitigate system degradation or disturbancepropagation. In some embodiments, for example, processing 100 furthercomprises automatically performing a prescribed action based on theindex. In other embodiments, processing 100 just comprises displayingthe index, e.g., to system operators that initiate control and/orprotective actions as they deemed appropriate.

In any event, regardless of the particular form of the index, computingthe index as described advantageously prevents or at least mitigatesfalse alarms in terms of prematurely detecting transmission corridorvoltage instability. This is because the Thevenin equivalent voltage andimpedance of the designated power generating part are tracked moreaccurately as compared to known approaches. Indeed, rather thanassigning any variation in the magnitude of the equivalent loadimpedance at the interface Int_(g) to the Thevenin equivalent impedance,embodiments herein selectively assign variations in that magnitude whichare deemed large to the Thevenin equivalent voltage instead. Such moreaccurately reflects the fact that these large variations areattributable to otherwise unmeasured changes in the designated powergenerating part (e.g., switching shunt capacitors on or off, generatorshitting their reactive power limits, line outages, etc.), asdistinguished from changes in local load.

FIG. 3 illustrates additional details about how the voltage stabilitymonitoring apparatus 8 tracks the Thevenin equivalent voltage andimpedance according to one or more embodiments. As shown, the voltagestability monitoring apparatus 8 models the power system 2 as includinga power generating part g (e.g., part A in FIG. 1), a power receiving(i.e., load) part l (e.g., part B in FIG. 1), and a transmissioncorridor 6 that interfaces part g at a power generating part interfaceInt_(g) and that interfaces part l at a power receiving part interfaceInt_(l). The apparatus 8 further models the power generating part g interms of its Thevenin equivalent voltage Ē_(g) and impedance Z _(g), andmodels the power receiving part l in terms of its equivalent loadimpedance Z _(l). The apparatus 8 tracks Ē_(g) and Z _(g) basedexclusively on a voltage phasor V _(Int) _(g) ^(i) and a current phasorĪ_(Int) _(g) ^(i) at interface Int_(g), and tracks Z _(l) basedexclusively on a voltage phasor V _(Int) _(l) ^(i) and a current phasorĪ_(Int) _(l) ^(i) at interface Int_(l).

FIG. 4 illustrates processing performed by the apparatus 8 to trackĒ_(g) and Z _(g) in this way, according to one or more embodiments. FIG.4's processing is performed for each of a plurality of phasormeasurement times i=0 . . . I; that is, for each time i a synchronizedmeasurement sample is received from the one or more PMUs 12 deployed atinterface Int_(g). A measurement sample received from a given PMU 12deployed at interface Int_(g) includes a voltage phasor V _(Int) _(g)_(,c) ^(i) measured at and a current phasor Ī_(Int) _(g) _(,c) ^(i)measured in transmission line c associated with interface Int_(g). Ofcourse Ē_(g) must be initialized at the start of FIG. 4's processing. Inat least some embodiments, for example, Ē_(g) is initialized to thearithmetic average of its extreme values:

$\begin{matrix}{{{{{\overset{\_}{E}}_{g}^{0} = \frac{{\overset{\_}{E}}_{g}^{{ma}\; x} - {\overset{\_}{E}}_{g}^{m\; i\; n}}{2}}{{where}\mspace{14mu}{\overset{\_}{E}}_{g}^{m\; i\; n}} = {{{\overset{\_}{V}}_{{Int}_{g}}^{0}\mspace{14mu}{and}\mspace{14mu}{\overset{\_}{E}}_{g}^{{ma}\; x}} = {{\overset{\_}{V}}_{{Int}_{g}}^{0}\frac{\cos\;\theta}{\cos\;\beta}}}},{{{where}\mspace{14mu}{\overset{\_}{Z}}_{l}} = {{Z_{l}\angle\;\theta\mspace{14mu}{and}\mspace{14mu}{\overset{\_}{E}}_{g}} = {E_{g}\angle\;\beta}}},{{and}\mspace{14mu}{where}}}\mspace{14mu}{{\tan\;\beta} = {\frac{{{\overset{\_}{Z}}_{l}^{0}{\overset{\_}{I}}_{{Int}_{l}}^{0}} + {{\overset{\_}{V}}_{{Int}_{l}}^{0}\sin\;\theta}}{{\overset{\_}{V}}_{{Int}_{l}}^{0}\cos\;\theta}.}}} & (1)\end{matrix}$

Regardless, as shown, the apparatus 8 computes a voltage phasor V _(int)_(g) ^(i) at interface Int_(g) for measurement time i (Block 210). Theapparatus computes V _(int) _(g) ^(i) in at least some embodiments as:

$\begin{matrix}{{\overset{\_}{V}}_{{Int}_{g}}^{i} = \frac{\sum\limits_{c \in {Int}_{g}}{P_{c}^{i}{\overset{\_}{V}}_{{Int}_{g},c}^{i}}}{\sum\limits_{c \in {Int}_{g}}P_{c}^{i}}} & (2)\end{matrix}$where P_(c) ^(i) is the power transfer through transmission line c atmeasurement time i. The apparatus 8 also computes an aggregated currentphasor Ī_(Int) _(g) ^(i) at interface Int_(g) for measurement time i(Block 220). In at least some embodiments, the apparatus 8 computesĪ_(Int) _(g) ^(i) as a function of the aggregated active and reactivepowers at interface Int_(g). In this case, the apparatus 8 computesĪ_(Int) _(g) ^(i) as:

$\begin{matrix}{{{\overset{\_}{I}}_{{Int}_{g}}^{i} = ( \frac{P_{{Int}_{g}}^{i} + {jQ}_{Intg}^{i}}{{\overset{\_}{V}}_{{Int}_{g}}^{i}} )^{*}}{where}} & (3) \\{P_{{Int}_{g}}^{i} = {\sum\limits_{c \in {Int}_{g}}P_{c}^{i}}} & (4) \\{Q_{{Int}_{g}}^{i} = {\sum\limits_{c \in {Int}_{g}}Q_{c}^{i}}} & (5)\end{matrix}$

Regardless, the apparatus 8 then computes an equivalent load impedance Z_(s) ^(i) at interface Int_(g) for measurement time i (Block 230). Asmodeled in FIG. 5, the equivalent load impedance Z _(s), is the load“seen” by interface Int_(g) (in the direction away from the powergenerating part g). Accordingly, for the current measurement time i, theapparatus 8 according to one or more embodiments computes the equivalentload impedance Z _(s) ^(i) at interface Int_(g) as:

$\begin{matrix}{{\overset{\_}{Z}}_{s}^{i} = \frac{{\overset{\_}{V}}_{{Int}_{g}}^{i}}{{\overset{\_}{I}}_{{Int}_{g}}^{i}}} & (6)\end{matrix}$

Having computed Z _(s) ^(i), the apparatus 8 tracks Ē_(g) and Z _(g) bycomputing Ē_(g) ^(i) and Z _(g) ^(i) for the current measurement time iin different ways depending on how much the magnitude of Z _(s) hasvaried since the previous measurement time i−1. Specifically, theapparatus 8 determines the size of the variation in the magnitude of Z_(s), as |Z _(s) ^(i)|−|Z _(s) ^(i-1)| (Block 240). The apparatus 8 isconfigured to deem the size of this variation as “large” if the size ofthe variation is greater than a threshold-defined variation. In at leastsome embodiments, this threshold-defined variation is a predefinedpercentage variation since the previous measurement time i−1, defined asε₁×|Z _(s) ^(i-1)|. As an example, the threshold ε₁ may have a value inthe range from 0.01 to 0.05. On the other hand, the apparatus 8 isconfigured to deem the size of the variation as “small” if the size ofthe variation is less than the threshold-defined variation. That said,in least some embodiments, the apparatus 8 deems particularly smallvariations as “insignificant” variations that do not justify updatingĒ_(g). Insignificant variations in this case are deemed to be variationsthat are less than a second threshold-defined variation. Such secondthreshold-defined variation may be a second predefined percentagevariation since the previous measurement time i−1, defined as ε₂×|Z _(s)^(i-1)|. As an example, the threshold ε₂ may have a value in the rangefrom 0.00005 to 0.001.

In at least some embodiments, the apparatus 8 dynamically adjusts thethreshold defining the first and/or the second threshold-definedvariation, as a function of the Thevenin equivalent voltage Ē_(g). Thatis, the apparatus 8 dynamically adjusts ε₁ and/or ε₂ as a function ofĒ_(g). In one embodiment, for example, the apparatus 8 dynamicallydecreases ε₁ and ε₂ responsive to increases in Ē_(g).

Irrespective of these details, if the apparatus 8 deems the size of thevariation in the magnitude of Z _(s) as “large”, the apparatus 8 adjuststhe Thevenin equivalent voltage computed for the previous phasormeasurement time (i.e., Ē_(g) ^(i-1)) by the magnitude of the change involtage phasor V _(int) _(g) at interface Int_(g) since that previousphasor measurement time (i.e., by |V _(Int) _(g) ^(i)−V _(Int) _(g)^(i-1)|) (Block 250). In at least some embodiments, the apparatus 8updates Ē_(g) in this way by updating the complex value Ē_(g) inrectangular coordinates (real and imaginary parts), as opposed toseparately updating the magnitude and angle of the complex value Ē_(g).In this case, the apparatus 8 computes the Thevenin equivalent voltageĒ_(g) ^(i) for the current phasor measurement time i as:Ē _(g) ^(i) =Ē _(g) ^(i-1)(1+|V _(Int) _(g) ^(i) −V _(Int) _(g)^(i-1)|)  (7)

By contrast, if the apparatus 8 deems the size of the variation in themagnitude of Z _(s) as “small”, the apparatus 8 increases or decreasesthe Thevenin equivalent voltage computed for the previous phasormeasurement time (i.e., Ē_(g) ^(i-1)) by a predefined percentage change(Block 260). In some embodiments, for instance, the predefinedpercentage change is |Ē_(s) ^(i-1)|×k|, where k is a pre-specifiedparameter configured to constrain tracking error within predefinedbounds. As an example, k may have a value in the range from 0.01 to0.0001, with k being set as higher within this range for a certainnumber of initial measurement sample times (e.g., until i=100) and beingset as lower within the range thereafter. Regardless, the apparatus 8decreases Ē_(g) ^(i-1) by the predefined percentage change when thevariation in the magnitude of Z _(s) (i.e., |Z _(s) ^(i)|−|Z _(s)^(i-1)|) has the same polarity as estimated variation in the Theveninequivalent impedance Z _(g) (i.e., |Z _(g) ^(i)*|−|Z _(g) ^(i-1)|, withZ _(g) ^(i)* being an estimate or intermediate evaluation of Z _(g) ^(i)that takes into account the present value of V _(Int) _(g) and Ī_(Int)_(g) and the previous value of Ē_(g)). Conversely, the apparatus 8increases Ē_(g) ^(i-1) by the predefined percentage change when |Z _(s)^(i)|−|Z _(s) ^(i-1)| does not have the same polarity as |Z _(g)^(i)*|−|Z _(g) ^(i-1)|.

Similarly to the case for largely sized variations, the apparatus 8 insome embodiments updates Ē_(g) responsive to small sized variations byupdating the complex value Ē_(g) rectangular coordinates (real andimaginary parts), thus directly providing correction for both themagnitude and angle of Ē_(g). This is contrasted with known approachesthat separately update the magnitude and angle of the complex valueĒ_(g). Updating the complex value Ē_(g) in rectangular coordinatesadvantageously allows the apparatus 8 to better track Ē_(g) in the faceof dynamic changes in the system. In one or more embodiments, forinstance, the apparatus 8 decreases Ē_(g) ^(i-1) by unconditionallycomputing Ē_(g) ^(i) for the current phasor measurement time i as:Ē _(g) ^(i) =Ē _(g) ^(i-1)(1−|Ē _(g) ^(i-1) ×k|)  (8)Likewise, the apparatus 8 increases Ē_(g) ^(i-1) by unconditionallycomputing Ē_(g) ^(i) for the current phasor measurement time i as:Ē _(g) ^(i) =Ē _(g) ^(i-1)(1−|Ē _(g) ^(i-1) ×k|)  (9)By updating Ē_(g) according to equations (8) and (9), the Theveninequivalent tracking technique herein advantageously applies to caseswhen active and reactive power flows over the corridor 6 have oppositedirections. This represents improvement over known Thevenin equivalenttracking approaches where such is not the case. For example, Corsi andTaranto's approach bound updates of Ē_(g) by a lower bound ε_(inf) andan upper bound ε_(sup). S. Corsi and G. N. Taranto, “A real-time voltageinstability identification algorithm based on local phasormeasurements,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1271-1279,August 2008. But the lower bound ε_(inf) is valid only if the active andreactive power flows have the same direction. And the upper boundε_(sup) is valid only at the voltage instability point (maximumdeliverable power). According to one or more embodiments, therefore,decreasing Ē_(g) ^(i-1) by unconditionally computing Ē_(g) ^(i)according to equation (8) and increasing Ē_(g) ^(i-1) by unconditionallycomputing Ē_(g) ^(i) according to equation (9) means that the apparatus8 does not condition the amount by which Ē_(g) ^(i-1) is decreased orincreased on the value of Corsi and Taranto's lower and upper boundsε_(inf), ε_(sup).

Finally, if the apparatus 8 deems the size of the variation in themagnitude of Z _(s) as “insignificant”, the apparatus 8 does not adjustthe Thevenin equivalent voltage Ē_(g) (Block 270). That is, theapparatus 8 simply computes the Thevenin equivalent voltage Ē_(g) ^(i)for the current phasor measurement time i as:Ē _(g) ^(i) =Ē _(g) ^(i-1)  (10)

Irrespective of how the apparatus 8 updates the Thevenin equivalentvoltage Ē_(g), the apparatus 8 then updates the Thevenin equivalentimpedance Z _(g) separately; that is, rather than jointly updating Ē_(g)and Z _(g) for the current measurement time i by simultaneously solvingfor Ē_(g) and Z _(g), the apparatus 8 first updates Ē_(g) withoutupdating Z _(g) and then updates Z _(g) based on the updated Ē_(g). Oneor more embodiments herein are indifferent to the method the Theveninequivalent impedance Z _(g) is updated. In embodiments shown in FIG. 4,though, the apparatus 8 updates the Thevenin equivalent impedance Z _(g)by solving a set of two linear equations for Z _(g), based on theThevenin equivalent voltage Ē_(g) (as updated/adjusted for the currentmeasurement time) (Block 280). Specifically, the apparatus 8 solves:

$\begin{matrix}{y_{i} = {H_{i}^{T}x_{i}}} & (11) \\{y_{i} = \begin{bmatrix}{V_{{Int}_{g},R}^{i} - E_{g,R}^{i}} \\{V_{{Int}_{g},I}^{i} - E_{g,I}^{i}}\end{bmatrix}} & (12) \\{x_{i} = \begin{bmatrix}R_{g}^{i} \\X_{g}^{i}\end{bmatrix}} & (13) \\{H_{i}^{T} = \begin{bmatrix}{- I_{{Int}_{g},R}^{i}} & I_{{Int}_{g},I}^{i} \\{- I_{{Int}_{g},I}^{i}} & {- I_{{Int}_{g},R}^{i}}\end{bmatrix}} & (14)\end{matrix}$where R_(g) ^(i)+jX_(g) ^(i) as the real and imaginary parts of Z _(g)^(i) are the two unknown variables for which the apparatus 8 solves theset of equations, and where E_(g,R) ^(i) and E_(g,I) ^(i) as the realand imaginary parts of E_(g) ^(i), V_(Int) _(g) _(,R) ^(i) and V_(Int)_(g) _(,I) ^(i) as the real and imaginary parts of V_(Int) _(g) ^(i),and P_(Int) _(g) _(,R) ^(i) and I_(Int) _(g) _(,I) ^(i) as the real andimaginary parts of I_(Int) _(g) ^(i) are known variables. ComputingThevenin equivalent impedance Z _(g) in this way does not involveassuming that equivalent reactive is negligible. Unlike Corsi andTaranto's approach which makes this assumption, therefore, the Theveninequivalent tracking approach herein advantageously extends to lowervoltage levels where this assumption does not hold.

Updating the Thevenin equivalent voltage Ē_(g) and impedance Z _(g) asdescribed above more accurately accounts for the impact that the powergenerating part g of the system 2 has on the stability conditions of thetransmission corridor 6. Indeed, changes in such stability conditionsare caused either by (A) changes in the power receiving (i.e., load)part l; or (B) changes in the power generating part g not directlymeasured by available measurements (e.g., switching shunt capacitors onor off, generators hitting their reactive power limits, line outages,etc.). If changes in the equivalent load impedance Z _(l) of the powerreceiving part l are only accompanied by changes in the current Ī_(Int)_(g) at the power generating part interface Int_(g), without largechanges in the voltage V _(Int) _(g) at that interface Int_(g), thosechanges should be account for with updates to the equivalent loadimpedance Z _(l). The Thevenin equivalent voltage Ē_(g) of the powergenerating part g should only be updated to reflect small changes in thesystem 2 (i.e., according to equations (8) and (9)). The apparatus 8detects that this scenario applies when the apparatus 8 detectsrelatively small variations in the magnitude of the equivalent loadimpedance Z _(s) at interface Int_(g). On the other hand, if changes inthe equivalent load impedance Z _(l) of the power receiving part l areaccompanied by changes in the current Ī_(Int) _(g) at the powergenerating part interface Int_(g), as well as by large changes in thevoltage V _(Int) _(g) at that interface Int_(g), those changes should beaccount for updates to the Thevenin equivalent voltage Ē_(g) of thepower generating part g (according to equation (7)). The apparatus 8detects that this scenario applies when the apparatus 8 detectsrelatively large variations in the magnitude of the equivalent loadimpedance Z _(s) at interface Int_(g).

Regardless of these additional details for tracking the Theveninequivalent voltage Ē_(g) and impedance Z _(g) of the power generatingpart g, the apparatus 8 herein of course computes an index indicatingthe voltage stability of the corridor 6 as a function of that trackedvoltage Ē_(g) and impedance Z _(g) (Block 130 of FIG. 2). In at leastsome embodiments, the apparatus 8 computes this index also as a functionof the T-equivalent of the corridor 6. FIG. 6 shows an example of suchembodiments.

In FIG. 6, the apparatus 8 computes the T-equivalent of the corridor 6based exclusively on the voltage phasor V _(Int) _(g) and current phasorĪ_(Int) _(g) at interface Int_(g) and on the voltage phasor V _(Int)_(l) and current phasor Ī_(Int) _(l) at interface Int_(l). Specifically,the apparatus 8 computes the T-equivalent represented in FIG. 6 by thecomplex impedances Z _(T) and Z _(sh) as:

$\begin{matrix}{{\overset{\_}{Z}}_{T} = {2( \frac{{\overset{\_}{V}}_{{Int}_{g}} - {\overset{\_}{V}}_{{Int}_{l}}}{{\overset{\_}{I}}_{{Int}_{g}} - {\overset{\_}{I}}_{{Int}_{l}}} )}} & (15) \\{{\overset{\_}{Z}}_{sh} = \frac{{{\overset{\_}{V}}_{{Int}_{g}}{\overset{\_}{I}}_{{Int}_{l}}} - {{\overset{\_}{V}}_{{Int}_{l}}{\overset{\_}{I}}_{{Int}_{g}}}}{( {\overset{\_}{I}}_{{Int}_{l}} )^{2} - ( {\overset{\_}{I}}_{{Int}_{g}} )^{2}}} & (16)\end{matrix}$Note that the apparatus 8 computes the T-equivalent in this way basedexclusively on phasor measurements for a current measurement time i,meaning that the computation is advantageously performed without delay.

Having computed the T-equivalent in this way, the apparatus 8 in theseembodiments proceeds by computing the Thevenin equivalent of thecombination of the power generating part g and the transmission corridor6, as shown in FIG. 7. Specifically in this regard, the apparatus 8computes the Thevenin equivalent impedance Z _(th) of the combinedgenerating part g and corridor 6 as:

$\begin{matrix}{{\overset{\_}{Z}}_{th} = {\frac{{\overset{\_}{Z}}_{T}}{2} + \frac{1}{\frac{1}{{\overset{\_}{Z}}_{sh}} + \frac{2}{{\overset{\_}{Z}}_{T} + {2{\overset{\_}{Z}}_{g}}}}}} & (17)\end{matrix}$The apparatus 6 then computes the Thevenin equivalent voltage Ē_(th) ofthe combined generating part g and corridor 6 as:

$\begin{matrix}{{\overset{\_}{E}}_{th} = {{\overset{\_}{Z}}_{{Int}_{l}}( \frac{{\overset{\_}{Z}}_{th} + {\overset{\_}{Z}}_{l}}{{\overset{\_}{Z}}_{l}} )}} & (18)\end{matrix}$where the equivalent load impedance Z _(l) of the power receiving (i.e.,load) part l is of course:

$\begin{matrix}{{\overset{\_}{Z}}_{l} = {- \frac{{\overset{\_}{V}}_{{Int}_{l}}}{{\overset{\_}{I}}_{{Int}_{l}}}}} & (19)\end{matrix}$

Regardless of the particular technique for calculating the Theveninequivalent of the combination of the power generating part g and thetransmission corridor 6, the apparatus 8 in at least some embodimentsuses this Thevenin equivalent in order to express the voltage stabilityindex directly or indirectly in terms of the system's maximumdeliverable power. The system's maximum deliverable power in this regardis reached when the magnitude of the Thevenin equivalent impedance Z_(th) of the combined generation and transmission parts of the system 2equals the magnitude of the equivalent load impedance Z _(l) of thepower receiving part: |Z _(Th)|=|Z _(l)|. The apparatus 8 is configuredto compute any stability index that exploits this relation, such as aratio of equivalent and load impedances, and reactive power margin(active, reactive, apparent).

One or more embodiments compute this voltage stability index accordingto the method of FIG. 2 in order to not only realize the advantagesdescribed above but to also more accurately account for structuralchanges within the transmission corridor 6. Specifically, the apparatus8 in such embodiments monitors whether a breaker for each line 4associated with the power generating part interface Int_(g) is open orclosed. The apparatus 8 exploits this breaker monitoring in order toimprove its monitoring of the equivalent load impedance Z _(s) at thepower generating part interface Int_(g). The apparatus 8 in this regarddynamically computes Z _(s) exclusively from phasor measurements takenat lines 4 whose breakers are closed.

Where the apparatus 8 computes Z _(s) according to equations (2)-(6),for example, the apparatus 8 excludes from consideration (andeffectively “removes” from the corridor 6) transmission lines cεInt_(g)that are flagged as having open breakers. In at least some embodiments,the apparatus 8 receives signals indicating the status of line breakersfrom phasor measurement units deployed for those lines and dynamicallyflags lines that have open breakers and that therefore should beexcluded from the corridor 6. Effectively “re-computing” the structureof the corridor 6 responsive to line breaker status in this wayadvantageously prevents zero-valued phasor measurements at open linesfrom introducing inaccuracies in computation of Z _(s).

In one or more embodiments, the apparatus 8 not only accounts forstructural changes within the corridor 6 that result in the opening of awhole transmission path, but also that result in the opening of a partof a transmission path. Consider the example shown in FIG. 8.

As shown in FIG. 8(a), the structure of the transmission corridor 6initially comprises the collection of three transmission paths 1, 2, and3 extending between virtual cuts 1 and 2 adjacent to physical buses 1and 2. Transmission path 1 consists of line 1A extending betweenphysical buses 1 and 5, line 1B extending between physical buses 4 and5, and line 1C extending between physical buses 4 and 2. Transmissionpath 2 just consists of line 2A extending between physical buses 1 and2. Finally, transmission path 3 consists of line 3A extending betweenphysical buses 1 and 3, and line 3B extending between physical buses 3and 2. Responsive to detecting that the breaker status for line 3A haschanged from closed to open, the apparatus 8 changes from computingP_(Int) ₁ as P_(1,1)+P_(1,2)−P_(1,3) to computing P_(Int) ₁ asP_(1,1)+P_(1,2). As shown in FIG. 8(b), though, the apparatus 8advantageously recognizes that the breaker status for line 3A changingto open only results in a partial opening of transmission path 3. Theapparatus 8 therefore adds the remaining portion of path 3 to bus 2 asinjection (here, as a positive injection given the direction of the flowof P_(2,3)). This means that the apparatus 8 continues to computeP_(Int2) as P_(2,1)−P_(2,2)−P_(2,3) both before and after detecting thechange of the breaker status for line 3A.

Intuitively, a change in the structure of the corridor 6 requiresre-initialization of the Thevenin equivalent voltage Ē_(g). However, oneor more embodiments herein refrain from re-initializing the Theveninequivalent voltage Ē_(g) in this case and instead rely on updates to theThevenin equivalent voltage Ē_(g) to accurately account for the corridorstructure changes. Specifically, responsive to detecting the opening orclosing of one or more breakers for lines associated with the powergenerating part interface Int_(g), the apparatus 8 simply updates theThevenin equivalent voltage Ē_(g) to reflect the magnitude of theresulting change in the voltage phasor V _(int) _(g) at interfaceInt_(g), as dynamically computed (to account for the corridor structurechanges), without re-initializing the Thevenin equivalent voltage Ē_(g).

Those skilled in the art will appreciate that the various embodimentsdescribed herein are presented as non-limiting examples. For instance,although FIG. 1 illustrates part A of the system 2 as being designatedas the power generating part g, those skilled in the art will appreciatethat such need not be the case. In fact, the apparatus 8 according toone or more embodiments dynamically adapts which of the parts of thesystem 2 is designated as the power generating part g. The apparatus 8does so responsive to detecting a change in the direction or magnitudeof power flowing through one or both interfaces between the corridor 6and the parts of the power system 2. In some embodiments, the apparatus8 monitors for such a change at every measurement time i.

The apparatus 8 dynamically detects a power flow direction change in oneor more embodiments as a function of the aggregated active power at thecorridor interfaces. Aggregated active power at an interface in thissense means the algebraic sum of active power over the transmissionlines associated with an interface. Consider the example shown in FIG.9.

As shown in FIG. 9(a), the apparatus 8 designates virtual bus 1 as thepower generating part interface Int_(g) responsive to detecting that theaggregated active power P₁ _(agg) at virtual bus 1 is positive and theaggregated active power P₂ _(agg) at virtual bus 2 is negative (wherepositive active power indicates power flow out of the bus and negativeactive power indicates power flow into the bus, as is conventional).Conversely, as shown in FIG. 9(b), the apparatus 8 designates virtualbus 2 as the power generating part interface Int_(g) responsive todetecting that the aggregated active power P₂ _(agg) at virtual bus 2 ispositive and the aggregated active power P₁ _(agg) at virtual bus 1 isnegative. Finally, as shown in FIG. 9(c), the apparatus designateswhichever virtual bus 1 or 2 has the larger aggregated active power P₁_(agg) or P₂ _(agg) responsive to detecting that both the aggregatedactive power P₁ _(agg) and P₂ _(agg) at buses 1 and 2 are positive. Insome embodiments, though, the apparatus 8 proceeds as shown in FIG. 9(c)only upon validating the positive polarity of both aggregated activepowers. Specifically, the apparatus validates that the phasormeasurements used for computing the aggregated active powers were notperformed over a short duration caused by fault conditions, as opposedto multi-terminal corridors with tapped configuration.

Of course, although the above embodiments have been described withreference to the direction and/or magnitude of active power flow, theembodiments apply equally to the direction and/or magnitude of reactivepower flow. In fact, in at least some embodiments, the direction ofactive power flow and the direction of reactive power flow are assumedto be the same.

Regardless, dynamically adapting the power generating part designationin this way advantageously generalizes voltage stability indexcomputation for application to a wide range of power system andtransmission corridor types. For instance, in some embodiments the powersystem 2 comprises an interconnected transmission system where the powerflow direction changes based on a series of operational decisions, suchas market rules, the schedule of power flow, and the demand-supply chainprocess. Of course, embodiments are also applicable to a radial system,where power statically flows from one part to another part of thesystem, but in such cases there is no requirement that one part bepreselected as the power generating part. Rather, changes in power flowdirection are handled through a series of recursive validation ofaggregated active powers at both ends of the corridor 6.

In one or more embodiments, the apparatus 8 re-initializes the Theveninequivalent voltage Ē_(g) responsive to dynamically adapting which partof the system 2 is designated as the power generating part g. Thisrequires a certain number of measurement samples to be used for Theveninequivalent voltage identification.

Advantages of improvements in voltage stability monitoring oftransmission corridors introduced by one or more embodiments herein areillustrated using phasor measurement units recordings for the exemplarycorridor shown in FIG. 10(a). Synchronized phasor measurements in thisexample are collected at the rate of 60 samples/second. The exemplarycorridor shown in FIG. 10(a) includes a line outage between buses 1 and3 at time t=60 seconds, with the same line re-closed at time t=192seconds. Advantages are illustrated in terms of two voltage stabilityindices derived from the concept of Thevenin equivalent: ratio ofequivalent and load impedances, as shown in FIG. 10(b), and reactivepower margin, as shown in FIG. 10(c). Time evolutions of these twoindices, with and without detection and incorporation of structuralchange within the corridor, are given in FIGS. 10(b) and 10(c). If theopened line is considered as part of the monitored corridor withmeasurement equal to zero (i.e., breaker status unknown), it produceserrors in corridor stability conditions computation of approximately 0.3in ratio of impedances and 400 Mvars in reactive power margin ascompared to the case when the opened line, together with its counterpartline at Cut 2, is taken out from the consideration in the algorithm(breaker status known).

A more complicated corridor is shown in FIGS. 11(a)-11(c) as anotherexample, to demonstrate the advantages of dynamically adapting whichpart of the power system is designated as the power generating part.This case includes consecutive outages of two lines at t=7 second andt=13 seconds, followed by reclosing the lines in opposite order ofoutages at t=20 seconds and t=25 seconds. Within the monitored corridor,these changes do not present structural changes, since they are notdirectly related to defined cuts of the corridor. Aggregated activepowers on the cuts are such that both are directed toward the corridor,with the power at Cut 1 having the bigger value and therefore Bus 1being designated as the power generating part. Computations introducedby one or more embodiments herein shown as being performed with correct(Bus 1 as power generating part) and wrong (Bus 2 intentionally set aspower generating part) polarity. Time evolutions of two major voltagestability indices are given in 11(b) (ratio of impedances) and 11(c)(reactive power margin). Wrong polarity results in considerable error:approximately 0.1 in ratio of impedances and 500 Mvars (biggest error)in reactive power margin.

Those skilled in the art will appreciate that the one or more “circuits”described herein may refer to a combination of analog and digitalcircuits, and/or one or more processors configured with software storedin memory and/or firmware stored in memory that, when executed by theone or more processors, perform as described herein. One or more ofthese processors, as well as the other digital hardware, may be includedin a single application-specific integrated circuit (ASIC), or severalprocessors and various digital hardware may be distributed among severalseparate components, whether individually packaged or assembled into asystem-on-a-chip (SoC).

Thus, those skilled in the art will recognize that the present inventionmay be carried out in other ways than those specifically set forthherein without departing from essential characteristics of theinvention. The present embodiments are thus to be considered in allrespects as illustrative and not restrictive, and all changes comingwithin the meaning and equivalency range of the appended claims areintended to be embraced therein.

What is claimed is:
 1. A method of monitoring voltage stability of atransmission corridor through which power flows between different partsof a power system, the method comprising the following performed by avoltage stability monitoring apparatus: monitoring an equivalent loadimpedance at an interface between the transmission corridor and a partof the power system designated as generating said power, the equivalentload impedance at said interface comprising a ratio of a voltage phasorat said interface to a current phasor at said interface; tracking aThevenin equivalent voltage and impedance of said designated part byseparately updating that voltage and impedance, wherein updating theThevenin equivalent voltage comprises updating the voltage to reflectthe magnitude of any changes in said voltage phasor that are associatedwith large variations in the magnitude of the equivalent load impedanceat said interface, said large variations including variations greaterthan a threshold-defined variation; and computing an index indicatingsaid voltage stability as a function of the tracked Thevenin equivalentvoltage and impedance.
 2. The method of claim 1, wherein said updatingthe Thevenin equivalent voltage comprises, for each of a plurality ofphasor measurement times, determining whether or not variation in themagnitude of the equivalent load impedance at said interface since aprevious phasor measurement time is greater than the threshold-definedvariation, and, if so, adjusting the Thevenin equivalent voltagecomputed for the previous phasor measurement time by the magnitude ofthe change in said voltage phasor since that previous phasor measurementtime.
 3. The method of claim 2, wherein said adjusting comprisescomputing the Thevenin equivalent voltage Ē_(g) ^(i) for a currentphasor measurement time i as Ē_(g) ^(i)=Ē_(g) ^(i-1)(1+|V _(Int) _(g)^(i)−V _(Int) _(g) ^(i-1)|), where Ē_(g) ^(i-1) is the Theveninequivalent voltage for a previous phasor measurement time i−1, V _(Int)_(g) ^(i) is said voltage phasor for the current phasor measurement timei, and V _(Int) _(g) ^(i-1) is said voltage phasor for the previousphasor measurement time i−1.
 4. The method of claim 1, wherein updatingthe Thevenin equivalent voltage further comprises, responsive to smallvariations in the magnitude of the equivalent load impedance at saidinterface, decreasing or increasing the Thevenin equivalent voltage by apredefined percentage change when said small variations do or do nothave the same polarity as estimated variations in said Theveninequivalent impedance, respectively, said small variations includingvariations less than the threshold-defined variation.
 5. The method ofclaim 4, wherein, responsive to small variations in the magnitude of theequivalent load impedance at said interface, increasing the Theveninequivalent voltage comprises computing the Thevenin equivalent voltageas Ē_(g) ^(i)=Ē_(g) ^(i-1)(1+|Ē_(g) ^(i-1)×k|) and decreasing theThevenin equivalent voltage comprises computing the Thevenin equivalentvoltage as Ē_(g) ^(i)=Ē_(g) ^(i-1)(1−|Ē_(g) ^(i-1)×k|), where Ē_(g) ^(i)is the Thevenin equivalent voltage for a current phasor measurement timei, Ē_(g) ^(i-1) is the Thevenin equivalent voltage for a previous phasormeasurement time i−1, and k is a pre-specified parameter configured toconstrain tracking error within predefined bounds.
 6. The method ofclaim 5, wherein, responsive to small variations in the magnitude of theequivalent load impedance at said interface, increasing the Theveninequivalent voltage comprises unconditionally computing the Theveninequivalent voltage as Ē_(g) ^(i)=Ē_(g) ^(i-1)(1+|Ē_(g) ^(i-1)×k|) anddecreasing the Thevenin equivalent voltage comprises unconditionallycomputing the Thevenin equivalent voltage as Ē_(g) ^(i)=Ē_(g)^(i-1)(1−|Ē_(g) ^(i-1)×k|).
 7. The method of claim 1, wherein updatingthe Thevenin equivalent voltage comprises updating the Theveninequivalent voltage's complex value in rectangular coordinates.
 8. Themethod of claim 1, wherein updating the Thevenin equivalent impedancecomprises solving a set of two linear equations with two unknownvariables that comprise the real and imaginary parts of the Theveninequivalent impedance, wherein known variables in the set of two linearequations include the real and imaginary parts of the Theveninequivalent voltage as updated to reflect the magnitude of any changes insaid voltage phasor.
 9. The method of claim 1, further comprisingdynamically adjusting a threshold defining the threshold-definedvariation, as a function of the Thevenin equivalent voltage.
 10. Themethod of claim 1, further comprising dynamically adapting which of saidparts of the power system is designated as generating said power,responsive to detecting a change in direction or magnitude of powerflowing through one or both interfaces between the transmission corridorand said parts of the power system.
 11. The method of claim 1, furthercomprising monitoring whether a breaker for each line associated withsaid interface is open or closed, and wherein monitoring the equivalentload impedance at said interface comprises dynamically computing theequivalent load impedance at said interface exclusively from phasormeasurements taken at lines whose breakers are closed.
 12. The method ofclaim 11, wherein, responsive to detecting the opening or closing of oneor more of said breakers, said updating comprises updating the Theveninequivalent voltage to reflect the magnitude of the resulting change insaid voltage phasor, as dynamically computed, without re-initializingthe Thevenin equivalent voltage.
 13. A voltage stability monitoringapparatus configured to monitor voltage stability of a transmissioncorridor through which power flows between different parts of a powersystem, the voltage stability monitoring apparatus comprising one ormore processing circuits configured to: monitor an equivalent loadimpedance at an interface between the transmission corridor and a partof the power system designated as generating said power, the equivalentload impedance at said interface comprising a ratio of a voltage phasorat said interface to a current phasor at said interface; track aThevenin equivalent voltage and impedance of said designated part byseparately updating that voltage and impedance, wherein updating theThevenin equivalent voltage comprises updating the voltage to reflectthe magnitude of any changes in said voltage phasor that are associatedwith large variations in the magnitude of the equivalent load impedanceat said interface, said large variations including variations greaterthan a threshold-defined variation; and compute an index indicating saidvoltage stability as a function of the tracked Thevenin equivalentvoltage and impedance.
 14. The voltage stability monitoring apparatus ofclaim 13, wherein the one or more processing circuits are configured toupdate the Thevenin equivalent voltage by, for each of a plurality ofphasor measurement times, determining whether or not variation in themagnitude of the equivalent load impedance at said interface since aprevious phasor measurement time is greater than the threshold-definedvariation, and, if so, adjusting the Thevenin equivalent voltagecomputed for the previous phasor measurement time by the magnitude ofthe change in said voltage phasor since that previous phasor measurementtime.
 15. The voltage stability monitoring apparatus of claim 14,wherein the one or more processing circuits are configured to adjust theThevenin equivalent voltage by computing the Thevenin equivalent voltageĒ_(g) ^(i) for a current phasor measurement time i as Ē_(g) ^(i)=Ē_(g)^(i-1)(1+|V _(Int) _(g) ^(i)−V _(Int) _(g) ^(i-1)|), where Ē_(g) ^(i-1)is the Thevenin equivalent voltage for a previous phasor measurementtime i−1, V _(Int) _(g) ^(i) is said voltage phasor for the currentphasor measurement time i, and V _(Int) _(g) ^(i-1) is said voltagephasor for the previous phasor measurement time i−1.
 16. The voltagestability monitoring apparatus of claim 13, wherein the one or moreprocessing circuits are configured to update the Thevenin equivalentvoltage also by, responsive to small variations in the magnitude of theequivalent load impedance at said interface, decreasing or increasingthe Thevenin equivalent voltage by a predefined percentage change whensaid small variations do or do not have the same polarity as estimatedvariations in said Thevenin equivalent impedance, respectively, saidsmall variations including variations less than the threshold-definedvariation.
 17. The voltage stability monitoring apparatus of claim 16,wherein the one or more processing circuits are configured, responsiveto small variations in the magnitude of the equivalent load impedance atsaid interface, to increase the Thevenin equivalent voltage by computingthe Thevenin equivalent voltage as Ē_(g) ^(i)=Ē_(g) ^(i-1)(1+|Ē_(g)^(i-1)×k|) and decrease the Thevenin equivalent voltage by computing theThevenin equivalent voltage as Ē_(g) ^(i)=Ē_(g) ^(i-1)(1−|Ē_(g)^(i-1)×k|) where Ē_(g) ^(i) is the Thevenin equivalent voltage for acurrent phasor measurement time i, Ē_(g) ^(i-1) is the Theveninequivalent voltage for a previous phasor measurement time i−1, and k isa pre-specified parameter configured to constrain tracking error withinpredefined bounds.
 18. The voltage stability monitoring apparatus ofclaim 17, wherein the one or more processing circuits are configured,responsive to small variations in the magnitude of the equivalent loadimpedance at said interface, to increase the Thevenin equivalent voltageby unconditionally computing the Thevenin equivalent voltage as Ē_(g)^(i)=Ē_(g) ^(i-1)(1+|Ē_(g) ^(i-1)×k|) and to decrease the Theveninequivalent voltage by unconditionally computing the Thevenin equivalentvoltage as Ē_(g) ^(i)=Ē_(g) ^(i-1)(1−|Ē_(g) ^(i-1)×k|).
 19. The voltagestability monitoring apparatus of claim 13, wherein the one or moreprocessing circuits are configured to update the Thevenin equivalentvoltage by updating the Thevenin equivalent voltage's complex value inrectangular coordinates.
 20. The voltage stability monitoring apparatusof claim 13, wherein the one or more processing circuits are configuredto update the Thevenin equivalent impedance by solving a set of twolinear equations with two unknown variables that comprise the real andimaginary parts of the Thevenin equivalent impedance, wherein knownvariables in the set of two linear equations include the real andimaginary parts of the Thevenin equivalent voltage as updated to reflectthe magnitude of any changes in said voltage phasor.
 21. The voltagestability monitoring apparatus of claim 13, wherein the one or moreprocessing circuits are further configured to dynamically adjust athreshold associated with the threshold-defined variation as a functionof the Thevenin equivalent voltage.
 22. The voltage stability monitoringapparatus of claim 13, wherein the one or more processing circuits arefurther configured to dynamically adapt which of said parts of the powersystem is designated as generating said power, responsive to detecting achange in direction or magnitude of power flowing through one or bothinterfaces between the transmission corridor and said parts of the powersystem.
 23. The voltage stability monitoring apparatus of claim 13,wherein the one or more processing circuits are further configured tomonitor whether a breaker for each line associated with said interfaceis open or closed, and are configured to monitor the equivalent loadimpedance at said interface by dynamically computing the equivalent loadimpedance at said interface exclusively from phasor measurements takenat lines whose breakers are closed.
 24. The voltage stability monitoringapparatus of claim 23, wherein the one or more processing circuits areconfigured, responsive to detecting the opening or closing of one ormore of said breakers, to update the Thevenin equivalent voltage toreflect the magnitude of the resulting change in said voltage phasor, asdynamically computed, without re-initializing the Thevenin equivalentvoltage.
 25. The method of claim 1, further comprising performing one ormore actions, based on the computed index, as needed to control thetransmission corridor's voltage stability and/or mitigate systemdegradation or disturbance propagation.
 26. The voltage stabilitymonitoring apparatus of claim 13, wherein the one or more processingcircuits are further configured to perform one or more actions, based onthe computed index, as needed to control the transmission corridor'svoltage stability and/or mitigate system degradation or disturbancepropagation.